Statistics on Z[X]: Some counting problems in the ring of univariate polynomials over the integers
Tamer Bulut (Universität Bonn)
Thursday 28 January 2010, 15.00, b-it 1.25 (cosec meeting room)
There are many useful algebraic and analytical analogies between the ring of integers and the ring of polynomials Q[X] over the rational numbers, e.g., the classical questions about the density of the set of prime numbers, the square-free integers, or the pairs of relatively coprime integers can also be formulated in suitable models for univariate polynomials over the integers. Some results to these counting problems will be discussed. For the polynomial ring Z[X] there arise even new questions: 1) What is the Galois group of a randomly chosen polynomial over the integers in a suitable model? The classical result of van der Waerden, that almost all integer polynomials have the full symmetric group as their Galois group, will be presented. 2) One can observe that there are monic irreducible polynomials over the integers, that are reducible modulo p for every prime p. How many of such polynomials are there in a canonical chosen model? An upper bound will be given.